While the cerebral cortex in the human brain is of functional importance, functions defined on this structure are difficult to analyze spatially due to its highly convoluted irregular geometry. This study developed a novel L1-norm regularization method using a newly proposed multi-resolution face-based wavelet method to estimate cortical electrical activities in electroencephalography (EEG) and magnetoencephalography (MEG) inverse problems. The proposed wavelets were developed based on multi-resolution models built from irregular cortical surface meshes, which were realized in this study too. The multi-resolution wavelet analysis was used to seek sparse representation of cortical current densities in transformed domains, which was expected due to the compressibility of wavelets, and evaluated using Monte Carlo simulations. The EEG/MEG inverse problems were solved with the use of the novel L1-norm regularization method exploring the sparseness in the wavelet domain. The inverse solutions obtained from the new method using MEG data were evaluated by Monte Carlo simulations too. The present results indicated that cortical current densities could be efficiently compressed using the proposed face-based wavelet method, which exhibited better performance than the vertex-based wavelet method. In both simulations and auditory experimental data analysis, the proposed L1-norm regularization method showed better source detection accuracy and less estimation errors than other two classic methods, i.e. weighted minimum norm (wMNE) and cortical low-resolution electromagnetic tomography (cLORETA). This study suggests that the L1-norm regularization method with the use of face-based wavelets is a promising tool for studying functional activations of the human brain.